Density and invariant means in left amenable semigroups

Abstract

A left cancellative and left amenable semigroup S satisfies the Strong Følner Condition. That is, given any finite subset H of S and any ϵ > 0 , there is a finite nonempty subset F of S such that for each s ∈ H , | s F △ F | < ϵ ⋅ | F | . This condition is useful in defining a very well behaved notion of density, which we call Følner density, via the notion of a left Følner net, that is a net 〈 F α 〉 α ∈ D of finite nonempty subsets of S such that for each s ∈ S , ( | s F α △ F α | ) / | F α | converges to 0. Motivated by a desire to show that this density behaves as it should on cartesian products, we were led to consider the set LIM 0 ( S ) which is the set of left invariant means which are weak ∗ limits in l ∞ ( S ) ∗ of left Følner nets. We show that the set of all left invariant means is the weak ∗ closure of the convex hull of LIM 0 ( S ) . (If S is a left amenable group, this is a relatively old result of C. Chou.) We obtain our desired density result as a corollary. We also show that the set of left invariant means on ( N , + ) is actually equal to LIM 0 ( N ) . We also derive some properties of the extreme points of the set of left invariant means on S, regarded as measures on βS, and investigate the algebraic implications of the assumption that there is a left invariant mean on S which is non-zero on some singleton subset of βS.